Hi, I have cycled nearly all of Zambia.
From Kapiri -> Lusaka will take you likely 3 days depending on your fitness level and how your children are doing. The roads get very busy once you leave Kabwe and head into Lusaka - although the other alternative is to take the dirt roads from Chisama into the city, although they are of varying conditions and your children again...
You'll be able to sleep in the bush without issue - there is ample Farmland in the area, and not any wild animals you need to really worry about. Come Lusaka it would be best you had a contact for a place to stay. Then its 500 some odd kilometres to Livingstone. Along the way there is a couple good climbs for you, but modereately easy, the road is in decent condition with wide shoulders. Try sleeping at the Police station before Mazabuko on the right hand side, There are two truck stops on the left hand side after you pass Choma, both will let you camp there.
Visa regulations are different per passport holder. Since I am Canadian it is $75 to cross into Zimbabwe. I enjoy Victoria falls on the Zambian side more.
Regards, Dave
tiredofit.ca